The Menger property on Cp(X,2)Cp(X,2)

A space X is said to have the Menger property (or simply X   is Menger) if for every sequence 〈Un:n∈ω〉〈Un:n∈ω〉 of open covers of X  , there exists a sequence of finite sets 〈Fn:n∈ω〉〈Fn:n∈ω〉 such that ⋃n∈ωFn⋃n∈ωFn is a cover of X   and Fn⊂UnFn⊂Un for every n∈ωn∈ω. We prove: (1) If X   is a subspace of Cp(Y)Cp(Y), where YnYn is Menger for every n∈ωn∈ω, and X′X′ (the set of non-isolated points of X  ) is compact, then Cp(X,2)Cp(X,2) is Menger; (2) If Cp(X,2)Cp(X,2) is Menger and X   is normal, then X′X′ is countably compact; (3) For a first countable GO-space without isolated points L  , Cp(L,2)Cp(L,2) is Menger if and only if Cp(L,2)Cp(L,2) is Lindelöf and L is countably compact; and for a subspace L   of an ordinal, Cp(L,2)Cp(L,2) is Menger if and only if Cp(L,2)Cp(L,2) is Lindelöf and L′L′ is countably compact; (4) For every F∈ω⁎F∈ω⁎, Cp(ω∪{F},2)Cp(ω∪{F},2) is Menger if and only if FF is a strong P  -point; (5) Assuming the Continuum Hypothesis, there is a maximal almost disjoint family AA for which the space Cp(Ψ(A),2)Cp(Ψ(A),2) is Menger.